Teaching Tuesday: Mathematical Journey - Part 1

Student math work

The teaching of math is fascinating. It is fascinating and complex. More than any of our other content areas, we have learned so much from our journey. A few years ago, at an MIT Club dinner, the gentleman next to me asked “Have you figured out how to teach problem solving?” My answer was quick, and I was excited to hear his response, “No, not yet, do you know how?” Alas he didn’t, and it led to a fun conversation about the teaching of math. Problem solving is at the core of math, and we’re still working on figuring out how to teach it to a broad range of students. Our experience has been illuminating, and a fun challenge to tackle.

When we designed the school, we read the research and work done by Patrick Suppes at Stanford University’s EPGY and by Julian Stanley at Johns Hopkins’ CTY to inform our approach that allowed students of varied math levels to progress at their own pace so that they could reach mathematically challenging material that would keep them engaged. Math, unlike other subjects taught in school (with the exception of foreign languages), builds on itself and requires a mastery of prior material to fully understand the new material being presented. In theory, self-paced math using adaptive technologies appears as a perfect solution. In practice, we found that it works for a smaller percentage of the population than we expected from the research, and even for those students for whom it works, they prefer the interaction with a person. 
The beauty of our school being a 4th-12th school is that it’s like a longitudinal study; You see how choices made in elementary years affect performance in high school. Here are some of our findings:

• Students’ trajectories in math vary. Some blossom in high school, while some race through arithmetic and “hit a wall” in high school.
• When working with larger groups, we are challenged to present math as an area to delve deeper into rather than a ladder to climb as quickly as possible.
• For some kids the “math world” comes together with calculus, when they see how it all fits together.
• In lower grades, topics learned in the math classroom don't readily transfer to the science classroom for all students.
• Some students have minds like steel traps that once the material is presented, that is all of the practice that they need, while others need a fair amount of repetition to master a concept.
• Confidence in math matters. Too fast, too soon can erode that confidence. Too much pressure to perform can also erode that confidence.
• When students make errors, analyzing if there is a commonality in the types of errors or response of the student is important in building self-awareness and confidence. 

 
For today, I will expand on this last bullet point, and we’ll visit other learnings in later posts. We have found some common areas of struggle in math which include computation errors, copying mistakes or not showing work, anxiety and freezing when working on a problem, and not understanding what the problem is asking. This is not an exhaustive list, and each could be the subject of its own post, but we’ll briefly explore each one.
 

• Computation errors: Most kids develop number sense and the math facts through the math curriculum by 6th grade. However, there is a minority of kids who I still see computing on their fingers in a high-school physics class, losing focus on the actual physics problem that they were trying to solve. Access to more advanced math and science classes was hindered, not because of an understanding of the physical concepts, but because they don’t have a sense of the order of magnitude to expect or spend so much time dealing with the numbers that they forget what they were solving for. Those multiplication tables do matter after all.
• Copying mistakes or not showing work: These kids tend to rush through the work to try to get through it as quickly as possible. In elementary and middle school math, it can work for them. However, by the time we get to mid-high school level math and science classes, a point here and there does add up. Kids tell me that it doesn’t matter, they understand the concept. I partially agree - yes, understanding the concept is important, and you can demonstrate that you know it by showing your work so you can get most of the credit. However, if you dream of a career as an engineer, financier, or doctor, mathematical precision matters. I want the engineers designing the plane I’m going to get on to be precise, and I want my anesthesiologist to be precise. Precision has to start somewhere. 
• Freezing when doing math: My trickiest students have been some of the students who teachers would normally describe as the “best” students in a class. In the elementary years, they take notes, they do their homework, they study, and they do well on assessments. They are well behaved, they smile, … and they generally don’t like to ask questions. All of the external signs they project in the classroom exude confidence and mastery, except for not asking questions. It’s not until they reach a certain math level, usually by Algebra 2, sometimes sooner, that we see the cracks. It’s like watching a duck on water, calmly on the pond but furiously treading water below the surface. For some kids it comes from an internal and/or external quest for perfection, for others it’s caused by their competitive drive to go faster, and yet for others, it’s because they compare themselves to their classmates. The causes are complex and working through it is difficult.
• Not understanding the problem: This is where reading meets math. The language in math textbooks corresponds to age-equivalent language. There is also vocabulary that is part of every field of study. We have seen a correlation between reading comprehension and the math classroom. 

​An observation in teaching that took me by surprise, that shouldn’t have, is that I always need more time. I feel that I could fill an entire year of school with just one area of exploration, and we could have great fun exploring. Next year, we are adding an extra math period for math that will give us time to explore additional topics not covered in elementary and middle-school math curricula, such as set theory and proofs. That extra time is also being built in so that students have an opportunity to explore how they learn math. See you next week as we continue to explore math!

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